Andrei Rodin (
[email protected]) Ecole Normale Supérieure
Identity and Categorification Content: 1. Paradoxes of Identity and Mathematics – p. 2 2. Mathematical Doubles – p.3 3. Types and Tokens – p.7 4.
Frege and Russell on The Identity of Natural Numbers – p.8
5.
Plato – p.11
6.
Equality and Equivalences in Geometry – p.12
7. Definitions by Abstraction – p.13 8. Relative Identity – p.14 9. Internal Relations – p.16 10. Classes – p.19 11. Individuals –p. 23 12. Relations versus Transformations – p.27 13. How to Think Circle? – p.33 14. Categorification – p.35 15. Identity in Categories: Overview – p. 42 16. Equality Relation and Identity Morphisms – p.44 17. Fibred Categories – p.48 18. Higher Categories – p.50 19. Platonic, Democritean, and Heraclitean Mathematics – p.56 20. Conclusion – p.63 Bibliography - p.65
1
1. Paradoxes of Identity and Mathematics Changing objects (of any nature) pose a difficulty for the metaphysically-minded logician known as the Paradox of Change . Suppose a green apple becomes red. If A denotes the apple when green, and B when it is red then A=B (it is the same thing) but the properties of A and B are different: they have a different color. This is at odds with the Indiscernibility of Identicals thesis according to which identical things have identical properties. A radical solution - to explain away and/or dispense with the notion of change altogether was first proposed by Zeno around 500 BC and remains popular among contemporary philosophers (who often appeal to the relativistic spacetime to justify Eleatic arguments). Unlike physics, mathematics appeared to provide support for the Eleatic position: for some reason people were more readily brought to accept the idea that mathematical objects did not change than to accept a similar claim about physical objects - in spite of the fact that mathematicians had always talked about variations, motions, transformations, operations and other process-like notions just as much as physicists. The Paradox of Change is the common ancestor of a family of paradoxes of identity which might be called temporal because all of them involve objects changing in time1. However time is not the only cause of troubles about identity: space is another. The Identity of Indiscernibles (the thesis dual to that of the Indiscernibility of Identicals) says that perfectly like things are identical. According to legend in order to demonstrate this latter thesis, Leibniz challenged a friend during a walk to find a counter-example among the leaves of a tree. Although there are apparently no perfect doubles among material objects, mathematics appears to provide clear instances immediately: think about two (different) points. But the example of geometrical space brings another problem: either the Identity of Indiscernibles thesis is false or our idea of perfect doubles like points is incoherent. In what follows I shall refer to this latter problem as the Paradox of Doubles. Mathematics looks more susceptible to this paradox than physics. However nowadays mathematics and physics are so closely entwined it is hardly possible to isolate difficulties in one
1
Chrisippus' Paradox, Stature, The Ship of Theseus belong to this family. See Deutsch (2002)
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discipline from those in the other. Were she living today, Leibniz's friend might meet his challenge by mentioning the indiscernibility of particles in Quantum Physics 2. Category theory provides an original understanding of identity in mathematics, which takes seriously the idea that mathematical objects are, generally speaking, variable and handles the problem of doubles in a novel way. Category theory does not resolve paradoxes of identity of the above form; rather it provides a setting where paradoxes in such a form do not arise. The Category-theoretic understanding of identity in mathematics may have important consequences in today's mathematically-laden physics and hence (assuming some form of scientific realism) for the notion of identity in a completely general philosophical setting. In this paper I explore this new understanding of identity in Category Theory, leaving its implications outside mathematics for a future study. The paper is organized as follows. First, I consider some difficulties about the notion of identity in mathematics, providing details and examples. Then I briefly review some attempts to overcome these difficulties. I pay particular attention to the account of identity in mathematics proposed by Frege and afterward developed by Russell, which remains standard in the eyes of many philosophers. Then I consider an alternate approach to identity in mathematics, which dates back to Greek geometry but made a new appearance in 19th century and later developed in Category theory. I consider the issue of identity in Category theory starting with general remarks and then coming to more specific questions concerning fibred categories and higher categories. Finally I suggest a way of thinking about categories, which implies deversification of the notion of identity and revision of Frege’s assumption that identities must be fixed from the outset.
2. Mathematical Doubles The example of two distinct points A,B (Fig.1) does not, it is usually argued, refute the Identity of Indiscernibles because the two points have different relational properties: in Fig. 1, A lies to the left of B but B does not lie to the left of itself3:
2
French (1988)
3
A•
•B
Fig.1
(The difference in the relational properties of A and B amounts to saying that the two points have different positions.) However the example can be easily modified to meet the argument. Consider two coincident points (Fig.2): now A and B have the same position.
A=B • Fig.2 It might be argued that coincident points are an exotic case, one which can and should be excluded from mathematics via its logical regimentation. But this is far from evident - at least if we are talking about classical Euclidean geometry. For one of the basic concepts of Euclidean geometry is congruence, and this notion (classically understood) presumes coincidence of points: figures F, G are congruent iff by moving G (without changing its shape and its size) one can make F and G coincide point by point. The fact that geometrical objects may coincide differentiates them significantly from material solids like chairs or Democritean atoms. The supposed impenetrability of material solids counts essentially in providing their identity conditions (Lucas 1973). Thus, identity works differently for material atoms and geometrical points4.
3
These relational properties of the two points depend on their shared space: the argument
doesn’t go through for points living on circle. I owe this remark to John Stachel. 4
This fact shows that Euclidean geometrical space cannot be viewed as a realistic model of the
space of everyday experience as is often assumed. One needs the third dimension of physical
4
We see that the alleged contradiction with the Identity of Indiscernibles is not the only difficulty involved here. Indeed the whole question of identity of points becomes unclear insofar as they are allowed to coincide. Looking at Fig.2 we have a surprising freedom in interpreting "=" sign. Reading "=" as identity we assume that A and B are two different names for the same thing. Otherwise we may read "=" as specifying a coincidence relation between the (different) points A and B. It is up to us to decide whether we have only one point here or a family of superposed points. The choice apparently has little or no mathematical sense. One may confuse coincidence with identity here without any risk of error in proofs. However this does not mean that one can just assimilate the notions of identity and coincidence. For identity so conceived would be very ill-behaved, allowing for the merger of different things into one and the splitting of one into many. (Consider the fact that Euclidean space allows for the coincidence of any point with any other through a suitable motion.) Perhaps it would be more natural to say instead that the relations of coincidence and identity while not identical in general, coincide in this context? For an example from another branch of elementary mathematics consider this equation: 3=3. Just as in the previous case there are different possible interpretations of the sign "=" here. One may read "=" either as identity, assuming that 3 is a unique object, or as a specific relation of equality which holds between different "doubles" (copies) of 3. Which option is preferable depends on a given context. There is a unique natural number x such that 22 is interesting because it no longer allows for thinking of different levels of the construction along the distinction between the “object level” and the “meta-level”. For when n is big the reiteration of “meta-” becomes useless and with n= ω it becomes senseless. So we cannot take refuge at the “meta-level» but must revise our understanding of identity from the outset. I cannot purport any technical discussion on higher categories in this paper but give the following simple example. Remind the “partial categorification» of an isomorphism class of sets which we have achieved in section 12: given such a class we considered all isomorphisms between its member-sets, then identified the sets and some (but not all) isomorphisms and got a symmetric group. Let it be finite symmetric group SN for simplicity. Now we can see that a more profound alternative to Fregean approach requires turning all sets into a category. Let us however for the sake of example improve on SN in a different way. Namely, let us categorify it further taking into account isomorphisms of SN, that is, group Aut(SN) of automorphisms of SN. Here we can remark something interesting. Except the trivial cases N=1 and N=2 when there exist only the identity authomorphisms, and except the “pathological” case N=6 we have Aut(SN) = SN 42. (This latter equality sign can be read as the isomorphism relation. Considering isomorphisms in question explicitly we get SN back!) So taking into consideration automorphisms of higher order (Aut(Aut(SN)) = Aut2(SN) and so on) brings nothing new: we have Autn(SN) = SN for all n and all N ≠ 1,2,6. Remark that SN equipped with Autk(SN), k=1,2,…n is a very simple albeit not completely trivial example of strict ncategory. The property of symmetric groups just mentioned is a case of what Baez&Dolan (1998) call stabilization in n-categories (p.13). Now we can fix some n, assume the “usual” equality only in Autn(SN) and for k
